Mathematical Sciences: Weighted Norm Inequalities and Partial Differential Equations

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This project is concerned with problems in two subareas of mathematical analysis: partial differential equations and harmonic analysis related to the Gaussian measure. The project has two objectives. The first is to analyze the behavior of solutions of a class of degenerate parabolic equations in divergence form with non-smooth coefficients. Such coefficients may be zero, infinite or both. The equations are known to model cases of diffusion of temperature in a non-homogeneous and non-isotropic material. Basic questions to be considered in this work include finding conditions which imply regularity of solutions, determining whether a Harnack principle is valid for non-negative solutions and understanding the behavior of the fundamental solutions of the equations. Weighted norm inequalities of the Poincare and Sobolev type will be used in this study. The second objective concerns the behavior, in the space of functions integrable with respect to Gaussian measure, of a class of transforms generated by the Ornstein-Uhlenbeck semigroup. These transforms play the role, in the Gaussian context, of the classical singular integrals of M. Riesz. Of particular interest will be the establishment of weak-type estimates with constants bounded independently of dimension. This investigation will center on the use of a maximal function as in the Calderon- Zygmund theory, but this function does not inherit all the properties of the classical maximal functions. However, there is evidence to suggest that it is still of weak type with respect to Gaussian measure. This is the first problem to be resolved.